**Solving absolute value equations on both sides :**

We may already know, how to solve absolute value equations in which we have absolute value sign on one of the sides, but not on both sides of the equation. In this section, we are going to see, how to solve the equations in which we have absolute value sign on both sides.

**Example 1 : **

Solve for x :

|x - 3| = |3x + 2|

**Solution : **

Based on the idea given above, we have

x - 3 = (3x + 2)

x - 3 = 3x + 2

- 5 = 2x

- 5/2 = x

(or)

x - 3 = - (3x + 2)

x - 3 = - 3x - 2

- 1 = - 4x

1/4 = x

**Justify and evaluation : **

Plug x = -5/2 and x = 1/4 in the given absolute value equation.

|-5/2 - 3| = |3(-5/2) + 2|

|-11/2| = |-15/2 + 2|

|-11/2| = |-11/2|

11/2 = 11/2

|1/4 - 3| = |3(1/4) + 2|

|-11/4| = |3/4 + 2|

|11/4| = |11/4|

11/4 = 11/4

Substituting x = -5/2 and x = 1/4 into the original equation results in true statements.

Both the answers x = -5/2 and x = 1/4 are correct and acceptable.

**Example 2 : **

Solve for x :

|x - 7| = |2x - 2|

**Solution : **

Based on the idea given above, we have

x - 7 = 2x - 2

x - 7 = 2x - 2

- 5 = x

(or)

x - 7 = - (2x - 2)

x - 7 = -2x + 2

3x = 9

x = 3

**Justify and evaluation : **

Plug x = -5 and x = 3 in the given absolute value equation.

**x = -5**

|-5 - 7| = |2(-5) - 2|

|-12| = |-10 - 2|

|-12| = |-12|

12 = 12

**x = 3**

|3 - 7| = |2(3) - 2|

|-4| = |6 - 2|

|-4| = |4|

4 = 4

Substituting x = -5 and x = 3 into the original equation results in true statements.

Both the answers x = -5 and x = 3 are correct and acceptable.

**Example 3 : **

Solve for z :

|2z + 5| = |2z - 1|

**Solution : **

Based on the idea given above, we have

2z + 5 = 2z - 1

5 = - 1

5 = -1

The above statement is false.

No solution here.

(or)

2z + 5 = - (2z - 1)

2z + 5 = -2z + 1

4 = -4z

-1 = z

**Justify and evaluation : **

Plug z = -1 in the given absolute value equation.

|2(-1) + 5| = |2(-1) - 1|

|-2 + 5| = |-2 - 1|

|3| = |-3|

3 = 3

Substituting z = -1 into the original equation results in true statement.

So, the answer z = -1 is correct and acceptable.

After having gone through the stuff given above, we hope that the students would have understood, "Solving absolute value equations on both sides".

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